Computer Science 789 Randomized Online Algorithms

Fall 2003

Assignments

This page will be updated frequently. Most recent update: December 9, 2003.

• Tuesday, August 26, 2003.
  Today we will look at a certain online problem, a version of the directed layered graph problem. There will be a worksheet on which each student attempts to find an optimal solution to each of three versions of the problem: offline, online, and randomized online. (The worksheets will not be handed in or graded.)
  We will discover that it is impossible to optimize the online case, in fact, the best that any online algorithm can do is to find a solution that costs twice as much as the optimal solution.
  We will then discover that if we are allowed to use randomization, we can do better, but still cannot expect to find the optimal solution.
  We will then briefly introduce the cache management problem, an online problem that will occupy a large portion of the course. In particular we will define LRU, a well-known optimal deterministic online algorithm for this problem, and briefly mention that the use of randomization can lead to performance better than LRU.
• Thursday, August 28, 2003.
  We will give the formal definitions of online problem and , followed by the formal definition of competitiveness.
  We will then give the formal definition of a randomized online algorithm, together with the definition of competitiveness in the randomized case.
  If time permits, we shall prove some lower bounds on competitiveness. In particular, we prove that the competitiveness of any deterministic online algorithm for the cache management problem, where the cache size is k, cannot be less than k.
  If time permits, we shall discuss the lower bound on the competitiveness of any randomized online algorithm for the same problem.
• Tuesday, September 2, 2003.
  Today we will discuss randomized algorithms for the cache problem.
  Random variables and expected value.
• Thursday, September 4, 2003.
  Today, we shall give a lower bound on the competitiveness of RANDOM.
• Tuesday, September 9, 2003.
  Assignment 1 is due in class today. The exercises must be hand written, not typed or printed from a computer file. Answers. Today we will give a simple randomized algorithm for the 2-cache problem that uses an unlimited number of bookmarks, and then prove that its competitiveness is optimal, that is, H₂ = 3/2. This algorithm uses unlimited memory. In fact, if the request sequence has length n, the memory requirement is O(n).
  We will then show how the amount of memory can be made finite, while retaining the competitiveness 3/2.
  If time permits, we will complete the proof that RANDOM is k-competitive for the k-cache problem.
• Thursday, September 11, 2003.
  Today we will continue the discussion from Tuesday. Read a discussion on potentials before class.
• Tuesday, September 16, 2003.
• Thursday, September 18, 2003.
• Tuesday, September 23, 2003. Assignment 2 is due in class today.
• Thursday, September 25, 2003. Read a discussion on the various models of randomized online algorithms before class.
• Tuesday, September 30, 2003. Read these files in this order:
  • Probabilities, Distributions, and Random Variables
  • Models of Online Algorithms
  • Using Potentials to Prove Competitiveness
  All three files have been substantially updated, and are in fact still under construction.
• Thursday, October 2, 2003. I will go over the handouts, with an emphasis on Models of Online Algorithms and Using Potentials to Prove Competitiveness.
• Tuesday, October 7, 2003. Doina will give you the solution to Exercise 2 of Models of Online Algorithms today.
• Thursday, October 9, 2003. Doina will help you with Assignment 3 in class today.
• Tuesday, October 14, 2003. Assignment 3 is due in class today.
• Thursday, October 16, 2003.
• Tuesday, October 21, 2003. We will introduce work functions. (Please let me know if you see any errors in that handout.)
• Thursday, October 23, 2003. I have found a large number of errors in the work functions handout. I put up a new version around 9:15, but there are still many errors. I hope to correct as many as possible in the next two hours or so, and will put up a running sequence of corrected versions so that you can give me feedback.
  I have stopped making corrections for today, at least until I pick up the printout around 3:30. If there are gross typos, I will correct them at that time.
• Tuesday, October 28, 2003. I have corrected some errors in the work functions handout and added new material.
  I have stopped making corrections for today, at least until I pick up the printout around 3:30. If there are gross typos, I will correct them at that time.
  Please print a copy of your own before class, and alert me to any errors you notice.
• Thursday, October 30, 2003. There are only a few differences between the current version and the version from Tuesday, but there are some unresolved issues, some raised by Dr. Bein during our research meeting yesterday afternoon.
  In any case, the proof of the competitiveness of K2 is complete.
  I have (unfortunately) not yet added exercises to the handout. This is very important, and I promise to do it over the weekend. Perhaps the class can suggest some.
  I made a few more corrections to the work functions handout based on errors pointed out to me in class this evening. Look for new versions tomorrow and Saturday.
• Friday, October 31, 2003. Dr. Bein and spent some hours going over the work functions handout this afternoon. There is one serious typo. which I will correct immediately. Look for substantial revisions over the weekend.
• Saturday, November 1, 2003. I have added a number of helpful exercises to the work functions handout, although I am still not finished adding the proofs that Dr. Bein and I worked out yesterday.
• Tuesday, November 4, 2003.
  There will be no more changes to work functions handout before class tonight. The last changes were at 2:17 this afternoon.
  I have corrected two minor errors pointed out by a student after class.
• Thursday, November 6, 2003. Normal office hour today.
  I will add to the work functions handout before class.
  We will discuss some of the problems in Assignment 4 in class today.
• Tuesday, November 11, 2003.
  Veterans' Day Recess.
  New names for the files:
  • Probabilities, Distributions, and Random Variables
  • Models of Online Algorithms
  • Using Potentials to Prove Competitiveness
  • Work Functions, Offsets, and Estimators
• Thursday, November 13, 2003.
• Tuesday, November 18, 2003. We will discuss more of the problems in Assignment 4 in class today, and some of them are to be handed in.
  Before we get to the server problem, I'll ask for comments/corrections/questions on Work Functions, Offsets, and Estimators, which has not been changed for several days. (I have not even fixed typos found recently, but will do so after class tonight or first thing tomorrow morning.)
  Today I will begin to discuss the server problem. There is, as yet, no handout for the server problem, but when I write one, it will be The Server Problem.
• Thursday, November 20, 2003. Today we will continue the discussion the server problem.
• Tuesday, November 25, 2003. Today we will continue the discussion the server problem. The server problem handout is now in its first stages. It contains a number of exercises. Try to work as many as you can before class today (do not hand them in) so that we can discuss them. I will write more in that handout this morning, to try to bring it up to at least where we ended the lecture last Thursday.
  As of Tue Nov 25 13:21:04 PST 2003, I have completed that task. Further changes this afternoon will be mostly to correct typos, which I expect you to help me with.
• Thursday, November 27, 2003.
  Thanksgiving Recess.
• Tuesday, December 2, 2003. Today there will be student evaluations of teaching performance, conducted by a staff member. I will delay coming to class so students can write their evaluations when I am not in the room. When the evaluations are done, someone can come to my office and tell me.
  We will discuss The server problem handout today.
• Thursday, December 4, 2003. I will try to finish the Final Examination Page today before class.
• Friday, December 5, 2003. I just fixed errors in the server problem handout. and in The work function handout. One of the errors was in the description of Action 21, a final exam problem, so that was important.
• Sunday, December 7, 2003. I made changes in the Final Examination Page today, some as late as Sun Dec 7 19:08:17 PST 2003.
  There is still a minor error in the figure for problem 6, even though I worked on it all day. But the error (I have p instead of r in one place) is explained in the Final Examination Page, and I will fix it tomorrow morning for sure.
• A major error in the work function handout was discovered by Qin Zhang. The corrected version is only very slightly different. The error was revealed trying to work the third problem on the final exam.
• Thursday, December 11, 2003. Final Examination, 6:00 PM.
• Monday, December 15, 2003. Well, it's over. To save disk space, I have erased all the pdf files, but left the source files publicly readable.